An elementary treatise on cubic and quartic curves, by A. B. Basset.

HARMONIC PROPERTIES.

63

and therefore
al =AD )sin~
% eAp-I 1 sin d'
whence
ao, o3 2a 1 2 \ sin
a7 7+3 _7 2 A P AR AQ) sin 0
= 0,
since AP is harmonically divided in P, Q and R.
If Q coincides with D, a2=0 and the theorem becomes
al,/7 + a3/Y3 = 0,
from which it follows that the four lines y, a - ky, a, a + ky form
a harmonic pencil. Also if four straight lines form a harmonic
pencil, any one of them is called the harmonic conjugate of the
other three.
101. Every line through a point of infiexion is divided
harmonically by the curve and the harmonic polar.
Let A be the point of inflexion; and let B and C be two of
the points in which the harmonic polar cuts the cubic. Then in
(5) we must put
ut = m/3 + ny, vl = 0, 3 = /3y (a/3 + v7),
and the equation of the cubic becomes
(m/3 + ny) a2 + 7y (3i + vey) = 0.............(14).
Let /3=k7y be any line through A which cuts the cubic in
P, and P3 and the harmonic polar BC in P2; substituting the
value of / in (14) it becomes
(ink + n) a7y + ky3 (kJ + v) = 0,
whence
al/7l + a3/3 = 0,
which shows that
1 1 2
AP1 + AP, AP2
102. Every chord drawn through a point on a cubic is cut
harmonically by the curve and the polar conic of that point.
Let AP1P2P3 be the chord cutting the cubic in A, P1, P'3and
the polar conic of A in P2. Then the equation of the cubic is